text:
How do these lines sound as ATC communications?
These are translated lines from a fictional story in Chinese, written by someone who (like me) does not work in aviation and therefore does not have firsthand knowledge of these things. Therefore, it is possible that the Chinese itself is an inaccurate portrayal, either of aviation in general or maybe of aviation in places other than China. If that seems to be the case, please just let me know.
I have put in bold font the parts of the lines below that I am most uncertain about.

(ATC) "[CALLSIGN], are you preparing to enter final? We’re changing runway directions. Approaching planes to the south of runway 17 left and right have all been pulled up due to wind shear."
(ATC) "[CALLSIGN], approved heading 350 as requested. And check your fuel levels. I’m directing you to holding point, it’s going to be a half-hour delay. Will that be a problem?"
(ATC) "[CALLSIGN], roger. Our radar… indicates it might take about half an hour for the weather to settle down."
(ATC) "[CALLSIGN], the first plane approaching on runway 04 has just landed. Do you want to turn around and take runway 04?"
(ATC) "How about this—I guide you to the north first, then south, fly the downwind leg, then land on 04?"
(ATC) "Correction, first turn to heading 360, north, then have you turn left and fly south. How’s that?"
(Pilot) "There’s weather from 5 to 10 nautical miles. It's not going to work."
(ATC) "Then what if you maintain your current heading, then go south… and finally turn to 180 for the approach?"
(Pilot) "How about I fly to the right for 20 nautical miles first, then make continuous left turns?"
(ATC) "[CALLSIGN], that’s possible, but you’re out of my zone if you continue for 20 nautical miles."
(ATC) "[CALLSIGN]—could you check the weather for me, 20 nautical miles at 10 to 11 o’clock?"
(Pilot) "From what I can see, it’s all yellow up here. Wind 300 at 32 knots."

For 3 and 6, I am unsure whether simply saying "weather" is too vague, but the Chinese does simply just use "有天氣," lit. there is weather.
For 5, the Chinese has "飛出個三邊來落04," which literally translates to something like "fly a downwind leg to land on 04." However, a consultant of mine told me that he couldn't really figure out what it means.
For 7, the Chinese has "連續左轉," lit. continuous left turns, turn left continuously; I'm not sure whether there is a more standard terminology that corresponds to that idea. And for "zone," I wasn't sure if I could use that to refer to an air traffic controller's area of jurisdiction (the ATC here is an approach controller).
For 8, based on what I've been able to find, I have a feeling it's not at all standard terminology in English. The Chinese is "本場上空是一大片黃區," lit. the air(space) above the current area is an expanse of yellow zone. I thought it might have to do with what this Quora response talks about, using the term "yellow area," but I'm not sure at all.
Edit: For 1, what I've translated as "wind shear" is "地面風的問題" lit. surface/ground? wind issues, but my consultant told me that it sounds very vague and I should just use wind shear, since elsewhere in the text it does clarify that the issue in question is wind shear. (My consultant is a non-native English speaker who lives in Taiwan and learned to fly in Australia, and now flies for fun occasionally.)
Here is the entire dialogue, which I may edit as I go. Feel free to let me know if the progression of events there don't make sense, and I can try to figure out whether that's an error in my translation or an inaccuracy in the original text. Directly commenting on the document is fine too. (But also feel free to ignore it; I am already asking for so much of your time.)
Thank you so much.

Broadly speaking those all make sense, but native English speakers might word things differently. Better aviation-related phraseology than you'll see in some media for sure! I'll see about making an answer if I have time. The discussion about "yellow areas" surely relates to precipitation as seen by aircraft-mounted radar (image included at this question).

@randomhead thank you!

It's hard to give a definitive answer to this, since there isn't standard phraseology for every possible situation. It's normal to use every day speech in situations like this, so it could sound in many different ways. This is my attempt at making it sound slightly more realistic:

"[CALLSIGN] are you ready for vectors to final? We're changing runway directions. We had multiple go-arounds on runway 17L and 17R due to wind-shear."
"[CALLSIGN] heading 350 approved. Expect holding and 30 minutes delay. Do you have enough fuel for that?"
"[CALLSIGN] roger. According to our radar we expect the weather to pass in approximately 30 minutes."
No change needed IMO.
"[CALLSIGN] can you accept vectors to the north and then back south for downwind for runway 04?"
No change needed IMO.
"Request right turn, continue 20 miles and then left hand orbits." / "That's possible, but you will leave controlled airspace if you continue for 20 miles."
No change needed IMO (yellow refers to the colour on the weather radar)

Re 3 & 6, I would say "weather" is sufficient. It's common to refer to areas of intense precipitation and turbulence (i.e. cumulonimbus clouds) simply as "weather".
Re 5, "downwind" is a standard term.
Re 7, "continuous left turns" could be the correct translation, depending on what they mean. My guess, however, is the meaning is either "left hand orbits" or "left hand holding". As for the "zone", it could either refer to the controllers area of responsibility, but in that case the controller could simply pick up the phone and coordinate with the adjacent sector - no real reason to involve the pilot in this decision. However, if they are referring to the limit of controlled airspace, it would make more sense to involve the pilot, since the pilot has to accept to leave controlled airspace.
As for 8, it seems weird that ATC asks the pilot to check the weather, when 3 seems to indicate that ATC has their own weather radar (not all ATC units do).

Thank you so so much! Based on the context from the rest of the story, I do think "zone" in 7 refers to the controller's area of responsibility - if that were the case, how might one phrase it? And is "miles" a shorter way to refer to nautical miles in these contexts?

(Also - I added one line to 6, but it's after you've already posted your answer, so please don't feel obligated to change anything!)

Excellent answer.  My only comment is on #2:  More typical I think is to either ask if you can accept a 30 minute delay, or straight up ask for fuel remaining, (in hours : minutes) if you have declared minimum fuel.  "Do you have enough fuel" just doesn't  seem like a question they'd ask.

@Pearl6527 In that case simply replace "controlled airspace" with "my area [of responsibility]". We always use nautical miles in aviation, so we usually just refer to it as "miles". #6 still sounds fine to me.

@MichaelHall Agreed, it sounds a bit awkward. I was trying to stay true to the origianl text. "Can you accept 30 minutes delay?" would be a better way of asking IMO.

Thank you both so much!! @MichaelHall (and 60levelchange, too, but I can't tag both of you)

@MichaelHall How might ATC ask for fuel remaining? (I.e. what would be a plausible way of phrasing the question?)

@Pearl6527 "Report fuel remaining"/"Report remaining fuel"

Yep... or possibly: "say fuel remaining"/"say fuel onboard"

@MichaelHall "Report" is standard phraseology, at least in Europe

OK, perhaps "say" is a product of my military background then...  But this these sort of contexts, "report" is used for future tense, while "say" is present tense.  For example, "report leaving 6000" vs "say altitude?"


text:
How might an ATC ask a pilot to check the weather conditions at a certain location?
(Working on a translation that involves radio communications dialogue.)
Say the location is X nautical miles from the aircraft to a direction that's between the Y o'clock and Z o'clock direction, what might an ATC say over radio to ask a pilot to check the weather conditions the location?
Sorry and please let me know if the information is given in a strange format—that's the way it's presented in the source-language text.
Edit: The original text is in Chinese and goes something along the lines of:

"[CALL SIGN]……您帮我看一下这儿天气，在您Y到Z点钟方位X海里这个地方。

for which a literal translation might go "[CALL SIGN], please check the weather here for me—X nautical miles at Y to Z o'clock."
The text is fictional and isn't meant to be technical, so it's possible that the phrasing doesn't match up with conventions. I just want to make sure I'm not translating it into something that's completely off-mark and would never be said in radio communications.
Thank you!

You mean to fly over and take a look?

What do you mean "that's the way it's presented in the text"?  What are you trying to "translate"?  Please provide source documents for a more complete answer, otherwise the short answer is for ATC to just ask for a PIREP.

@MichaelHall I'm sorry for being vague in my question. I've edited to provide more information. The source text is a fictional story and not any sort of technical document.

Much better, thanks.  Your literal translation is actually pretty good as-is.  I wouldn't quibble over it if I read it in a novel.  Sometimes radio communication is simply plain language, especially in slightly non-standard scenarios.

@MichaelHall Got it—thank you!

"Check the weather" is pretty vague. Is the controller asking about visibility, cloud base, precipitation, icing conditions, turbulence or something else entirely? It seems unrealistic that the controller does not specify exactly what information he is looing for.

Would ATC ever use “X o’clock” when communicating with an aircraft (except maybe while taxiing)? Wouldn’t they always use headings instead?

A controller would never ask an aircraft to go out of its way to check on weather, except in some emergency where the information is needed to help the other airplane with the emergency. Otherwise, it would only apply where the controller as asking for a pilot's observation from where they are, which is done from time to time.
A scenario might be, a VFR airplane stuck on top of a cloud layer, calls ATC for help, and a controller trying to help the pilot get down might ask an aircraft in the vicinity with a good vantage point if the area around location X is in the clear, so the controller could direct the VFR pilot to that area to get down.
If the situation is dire enough and some distance has to be covered to make the observation, the controller might even ask the pilot, if able, to divert to check it out, and it would be up to the pilot to do so.

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text:
How might ATC phrase an instruction to a pilot to maintain a certain speed and establish glide path?
What might an ATC say to instruct a pilot to maintain a certain speed and establish glide path?
My attempts (as someone who does not know much at all about how a plane is piloted) are:

"Maintain XXX onto glide path"
"Maintain XXX, report established on glide path"
"Maintain XXX until intercepting glide path"

but I'm not sure about them at all, nor about any potential differences in nuance between them.
Thank you!

I'm sure one of the ATC folks will give a better answer based on standard phraseology, but #3 is closest...

ICAO standard phraseologies are  given in PANS-ATM Doc 44444 16th edition.
Refer the following standard Phraseologies given in the 12th Chapter.
Para 12.4.2.2

d) CLEARED FOR (type of approach) APPROACH RUNWAY (number);
k) MAINTAIN (altitude) UNTIL GLIDE PATH INTERCEPTION;

e) REPORT ESTABLISHED ON LOCALIZER (or ON [GLS/RNP/MLS] [FINAL] APPROACH [COURSE]);

Para 12.4.1.6

c) MAINTAIN (number) KILOMETRES PER HOUR (or KNOTS) [OR GREATER (or OR LESS)] [UNTIL (significant point)];

So these blocks of phraseologies can be adapted  to the above situation as

CLEARED FOR (Type of Approach) Runway XX. MAINTAIN 190 KNOTS UNTIL GLIDE PATH INTERCEPTION.[REPORT ESTABLISHED]

Note that contents inside the square brackets in the sample phraseology are optional

Thank you so much!!

Is GLS/glideslope more commonly used than glide path? (Or do they not refer to the same thing?)

Sorry—did a search and realized GLS doesn't refer to glideslope. In that case, is "report established on glide path" not something that's said?

@Pearl6527 that's really depends on how the aircraft is carrying out the approach. Whether it is vectored on to the pilot interpreted final approach or carrying out a complete instrument approach etc. all the following are standard phraseologies depending upon the situation. "Report Established on Localizer", "Report established on ILS" , " Report establed on glide slope"

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text:
Could someone help me make this checklist-execution dialogue sound correct?
I am translating from Chinese a dialogue that involves two pilots going through an approach checklist. However, all the information I could find on the internet seemed to be for flight simulators (I think?) and I'm really lost.
Below is what I currently have, and I'm pretty sure there's more wrong than right...
In any case, the speaker alternates every line (i.e. lines 1, 3, 5... = pilot A, lines 2, 4, 6... = pilot B).

“Approach procedure.” ??
“Approach procedure confirmed.”
“Approach setup.” ??
“Approach setup complete.”
“Speed.”
“220 KIAS.” (simply "knots"?)
“EICAS.”
“Checked.”
“Landing lights.”
“On.”
“Spoilers.”
“Armed.”
“Autobrakes.”
“Set.”
“Flaps.”
“Five degrees.”

And the original Chinese goes:

And a link to a document with the text content if it's at all helpful: https://docs.google.com/document/d/1Q4TmYohNdlmav4DtZmKIazRNP7j5qjIjCnJ0kLboxFQ/edit?usp=sharing
The Chinese text is from a fictional story, and there is a possibility that the Chinese itself is either not representative of how approach checklists are done or only representative of how they do the checklists in China.
Thank you!

Are you being paid to translate word for word, or do you have creative license to condense at all?  For example:  "The pilots completed the approach checklist.". (I know you aren't asking for writing advice, but...)

@MichaelHall I don't have to translate it word for word, but I would ideally like to have a somewhat similar dialogue structure to the original text (so like roughly line for line, but the contents of the lines don't have to match up exactly).

I am thinking, though, that I could for example do away with the first four lines if they really don't correspond to or mean anything.

Taking the Descent Checklist from this site, which, as a checklist, looks about like this:

DESCENT CHECKLIST
Pressurisation . . . . . . . . . . . . . . . . . . . . . . . . . . . .LAND ALT___
Recall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Checked
Autobrake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ___
Landing data . . . . . . . . . . . . . . . . . . . .VREF___, Minimums___
Approach briefing . . . . . . . . . . . . . . . . . . . . . . . . . . . Completed

This would sound something like this:

"Descent Checklist"
"Descent Checklist: Pressurization"
"Land Altitude 1,000 feet, set"
"Recall"
"Checked"
"Autobrake"
"Off"
"Landing data"
"V-Ref 142, minimums 1,200 feet, set"
"Approach Briefing"
"Completed"
"Descent Checklist complete."

The first line would be the pilot flying (PF) calling for the Descent Checklist; the second line would be the pilot monitoring (PM) responding with the name of the checklist and reading the first challenge. The PF responds, the PM reads the next challenge, and so on until after the response to the last item (approach briefing) has been given, the PM announces that the checklist is complete.
The "VRef" challenge is for the landing speed, which typically is just given as the numeric value and without any "knots" required... we all know what units our speed is in.  Which does make the "feet" part of the Pressurization and Minimums lines seem extraneous, but for whatever reason those seem to get mentioned more than "knots" do. But you could include all the units or exclude them all without damaging the realism significantly.

Thank you! Do you know when/if landing lights get checked? Because landing lights end up playing a role in a subsequent part of the story, and I would like to somehow mention them here if possible.

During approach landing lights are typically set on when decending through 10 000ft. Depending on SOP the light status is checked once or more times during the rest of the approach, but they should be on below 10 000ft, night and day.

Perfectly reasonable to insert "Landing lights" "On"  as another item in this checklist.  If you want to, you could also add dialog that indicates when, in terms of altitude in the descent, the checklist is being run. The logical way to do that would be the PF announces "Flight level 180, Descent Checklist" and the first challenge/response becomes "Altimeters" "30.12 set". That way, it's clear the checklist is being run descending through 18,000', which is a perfectly appropriate point to have the landing lights on.


text:
Do these phrases sound like actual things that could be said during radio communications between pilots and ATCs?
I'm working on some translations and want to make sure that I'm not making things up with the radio communications.
Here are a few sentences I would appreciate help with. Bold font indicates areas I'm particularly unsure about (for example, places where I'm not completely sure what the original text means, and am not sure how best to translate).

[FLIGHT #], taxi out, runway 2R, turn right heading 090 for takeoff.
[FLIGHT #], radar contact, ETUDE-10 for takeoff, climb and maintain at 2200 meters, QNH 998.
[FLIGHT #], radar contact, VAK01 approach, runway 17L, descend and maintain at 3600.
Maintain heading 030 at 1500 height. Please confirm intentions.

Also, are there any recommendations for resources on this? I know there are recordings of ATC communications, but are there any written sources such as transcripts?
Thank you for any help!
Edit: Here are some ATC recordings with captions I found for airports in China -

https://www.bilibili.com/video/BV1Cs411V7gt/?spm_id_from=333.788.recommend_more_video.0&vd_source=6573b5aa3eb27ecb2432df9114663b88 (Shanghai Approach, see for example at around 1:27)
https://www.bilibili.com/video/BV1MW411g7D2/?spm_id_from=333.788.recommend_more_video.4&vd_source=6573b5aa3eb27ecb2432df9114663b88 (Beijing Approach, e.g. around 4:58)

Are ATC communications in China in a different format from those most commonly used elsewhere?

It would help if you provided the country so that a reader can compare it to required phraseology. In the US we don't use meters, but maybe other countries do.

None of those sound remotely like ATC communication in the US.  Somewhere else, maybe.

@RetiredATC
These are translations of what I believe were intended to be ATC communications in China.

In the US, we would generally use “altitude” in lieu of “height” but that may be an artifact of your translation.  In most ATC communications the world “altitude” isn’t even used, “At or below five thousand” is implicitly altitude.  Also, QNH has not been used in the US prbably since the time when navigator were last sent route information by Morse code.  There may be some countries still using it in lieu of “Altimeter.”  And in the US, altitudes would be feet, not meters.

If your goal is to capture the technical details of the communication, (I.e. content)it is likely you have succeeded already.  If for some reason you want to make a perfect conversion over to standard phraseology in the US, then you have more work to do.

@Max So would something like "altimeter 998" be used instead of "QNH 998"?

For example, there's some communication in English in this captioned recording (https://www.bilibili.com/video/BV1MW411g7D2/?spm_id_from=333.788.recommend_more_video.4&vd_source=6573b5aa3eb27ecb2432df9114663b88) of the approach controller in Beijing Capital Airport, and at around 4:58 there's "Asiana 333, descend and maintain 2400m, QNH 1028."

@Pearl6527 in the US the altimeter setting would be expressed in inches of mercury, not millibars.  998mb is 29.47 in HG, so in the US you’d here something like “Minneapolis altimeter 29.47”.

[FLIGHT #], taxi out, runway 2R, turn right heading 090 for takeoff.

This one is a little confusing. I believe this is intended to be a clearance for takeoff. The "taxi out" part isn't standard phraseology, but if I heard it, I would interpret it as instructions to "line up and wait" (i.e. go out on the runway, but not take off yet).* On the other hand, you're never supposed to use the word "takeoff" except in an actual clearance for takeoff. Also, you'd never give a heading instruction until the plane is actually ready to go.
If ATC wasn't intending for the plane to take off just yet, I would expect something like "[FLIGHT #], runway 2R, line up and wait" or "[FLIGHT #], runway 2R, position and hold" (depending on the country).
If ATC did want the plane to take off, I would expect "[FLIGHT #], right turn heading 090, Runway 2R, cleared for takeoff." The turn instructions mean that, after takeoff and once the plane has reached a safe altitude, turn right to 090 (due East).

[FLIGHT #], radar contact, ETUDE-10 for takeoff, climb and maintain at 2200 meters, QNH 998.

Aside from the "ETUDE-10 for takeoff", this is a standard phrase when a plane has just taken off, and is contacting departure control. They want the plane to climb to 2200 meters above mean sea level, and the local altimeter setting is 998.
I have no idea what "ETUDE-10 for takeoff" means. "ETUDE-10" sounds like it might be a specific departure procedure ATC wants the plane to fly, but I have no idea why they'd put "for takeoff" when the plane is already in the air at this point. Is it possible that the controllers are using the same word for "takeoff" as "departure"?

[FLIGHT #], radar contact, VAK01 approach, runway 17L, descend and maintain at 3600.

This one sounds like a pretty normal communication for a plane approaching an airport, and contacting approach control. "VAK01 approach" would be the name of the specific approach procedure that ATC wants the plane to fly. Although, I would expect a few more words, such as "fly the VAK01 approach" and "expect runway 17L".

Maintain heading 030 at 1500 height. Please confirm intentions.

This is the most normal-sounding of the four. The word "height" would normally be left off, or it would be translated "at a height of 1500". The bold section refers to a height of 1500 meters** above mean sea level.
* Well, okay, that's not exactly true. If I heard that, I would actually ask for clarification. But I'm trying to interpret this the best I can without being able to ask.
** I'm assuming meters, based on the fact that translation #2 used them, and 1500 feet would be an incredibly low altitude for most aircraft.

Thank you so much! For the ETUDE-10 one, it is definitely possible that the word should be translated as "departure" instead of "takeoff." Would something like "ETUDE-10 for departure" make more sense?

@PinyuHwang I wouldn't expect the word "for" to be in there. The full phrase is "Fly the ETUDE-10 departure", and if the controller is rushing, they might drop the first two words, leaving just "ETUDE-10 departure".

Okay I see. Thank you!

NTSB accident reports typically include printed transcripts of the CVR recordings, which include edited transcripts of the radio traffic.  While these transcripts aren't comprehensive (they omit lots of radio traffic that doesn't concern the accident aircraft and doesn't shed any light on the accident), they give a good flavor of what actual radio traffic typically sounds like.
The limitation is, nearly all of the transcripts are from US ATC.  Accident reports from other countries may contain similar transcripts that will give more flavor for their particular region.
As noted in my comment, none of the statements posted in the OP read much like any sort of ATC traffic in the US.  For another region, they may be much closer.

Hmm I see. Thank you for the pointer!

It’s doubtful any other jurisdiction would say these things either.  Several are just completely out of order and the rest don’t use recognized phraseology.


text:
What would an ATC command telling a pilot to maintain a certain speed until a certain distance from the destination sound like?
For example, instructing an aircraft to maintain a speed of 200 knots until it is 10 nautical miles from the destination / runway threshold?

In the united-states it would sound something like this, in the absence of an otherwise-useful named fix/waypoint/NAVAID:

Citation 3AB maintain two zero zero knots until ten-mile final.

You are more likely to hear "five-mile final" as that is the latest point ATC is allowed to require the pilot to fly a specific speed, as spelled out in FAA JO 7110.65 5–7–1:

b. Do not assign speed adjustment to aircraft: 4. Inside the final approach fix on final or a point 5 miles from the runway, whichever is closer to the runway.

Shrug. We say that kind of thing with some regularity—not speed control necessarily (although sometimes) but more often for sequencing arrivals: "Enter right base for a three-mile final." "Cleared visual approach, join no closer than a five-mile final." I think it's perfectly acceptable to let pilots do pilot things to estimate their position with respect to a runway, if we're using single-digit numbers of miles (and I don't expect to-the-foot accuracy either).


text:
Formula relating propeller torque to lift
I'm trying to find a formulas that relates lift to torque on the rotor shaft. Torque, angular velocity and power are related by the following relationship: $P=\tau \omega$.
I am especially interested in the relationship between the torque on the propeller shaft and the lifting force (not thrust). And how the vectors of these forces change when the position of the propeller in space changes.
I.e., I try to find something like $Lifting Force = f(\tau, attitude, \omega, P, etc.)$
There are some tips in this thread, but it's related to thrust, not lift: Are there equations to match power required to mass for rotorcraft?

This is a rather vague question to a rather complex topic. To answer it adequately, several chapters of a book would be needed. Could you please narrow the scope?

@PeterKämpf I can, but then I’ll have to partially return to the previous question. What I'm really interested in first is the relation between propeller torque (or power) to lifting force. See my edit please.

Since the topics are "helicopter" and "quadcopter", how do you differentiate lift from thrust?

@PeterKämpf I am an amateur in aviation. In my understanding, the forces when changing the position of the propeller change as follows. I want to know the relationship between the lift force and the moment on the propeller shaft. https://ibb.co/RgTWRDC That is, in the vertical case, these forces coincide, but when the tilt changes, the total force breaks down into projections.

Wouldn't you agree that it is easier to equate both? Lift/thrust sits perpendicular on the rotor disk (assuming no wind!) and this times sine of the disk inclination can then be considered as horizontal thrust component. If you keep that strange definition, you will have lots of problems, like: When would you do the transition between hover and flight?

@PeterKämpf as I understand it: if the rotor disk is horizontal, then there is only lifting force. To make an aircraft with a propeller move "forward", we need to tilt the rotor disk slightly (for example, copters or helicopters fly tilted slightly forward). At this point, in addition to lifting force, there is also thrust. But this does not change the fact that between the torque on the rotor shaft and the forces (or projections of forces) there must be a relationship that I am looking for. Of course, that this be a convenient tool.

Gliders use the lift of their wings for acceleration and deceleration all the time ... nobody thinks of calling this "thrust".

@PeterKämpf I'm talking about something that is even reflected in these drawings https://cdn.shopify.com/s/files/1/2773/1296/files/How_Do_Helicopters_Fly_-_Pilot_Mall_9481f40e-c097-4a4e-af60-1b8804297fe9.png?v=1702573248 https://media.hswstatic.com/eyJidWNrZXQiOiJjb250ZW50Lmhzd3N0YXRpYy5jb20iLCJrZXkiOiJnaWZcL2hlbGljb3B0ZXItZmx5LmdpZiIsImVkaXRzIjp7InJlc2l6ZSI6eyJ3aWR0aCI6MjkwfX19

@PeterKämpf so, can you tell me the formulas that connect "lift", "thrust" and "torque"?

First of all, lift is normally defined as the aerodynamic force component orthogonal to the direction of airflow (which equals the direction of movement in still air) while thrust is reserved for propulsion systems and denotes the resulting force. For helicopters or multicopters, the lift should be equal to the thrust component which is orthogonal to the direction of movement.
The power absorbed by a prop is equal to torque multiplied by angular velocity or thrust multiplied by speed and divided by propeller efficiency.  Here, speed is measured parallel to thrust, so for an airplane propeller this would be flight speed. For helicopters, it should be the speed component through the rotor parallel to thrust. To compute torque in terms of lift, you now need to know the angle between thrust and lift. If we assume horizontal motion in still air, this angle depends on the drag caused by this horizontal motion.
Static hover
Now we need to differentiate between hover and forward flight. Lift equals thrust in hover, but since forward speed is zero, we need a different equation to connect it with power:
$$T_0 = \frac{P\cdot\eta_{Prop}}{\sqrt{\frac{2\cdot T_0}{\pi\cdot d_P^2\cdot\rho}}} = \sqrt[\LARGE{3\:}]{P^2\cdot\eta_{Prop}^2\cdot\pi\cdot \frac{d_P^2}{2}\cdot\rho}$$
What is now missing is the rotational speed of the rotor. If we call this $\omega$, the torque becomes:
$$\tau = \frac{T_0}{\omega} = \frac{\sqrt[\LARGE{3\:}]{P^2\cdot\eta_{Prop}^2\cdot\pi\cdot \frac{d_P^2}{2}\cdot\rho}}{\omega} = \frac{L}{\omega}$$
Forward flight
For an airplane propeller thrust and speed align, so we can write:
$$\tau = \frac{P}{v\cdot\omega}$$
For helicopters we first need to work out the direction of thrust (which depends on flight speed, as drag goes up with its square) and the flow speed through the rotor. This is not trivial and I recommend to write some software following the math explained in the books recommended by @Sophit.
Nomenclature:
$\kern4mm d_p\kern6mm$propeller diameter in m
$\kern4mm L\kern7.5mm$lift force in N
$\kern4mm P\kern7.5mm$Power in Watt
$\kern4mm T\kern7.5mm$thrust in N
$\kern4mm T_0\kern5.5mm$static thrust in N
$\kern4mm v\kern8mm$flow speed
$\kern4mm \eta_{Prop}\kern1mm$propeller efficiency
$\kern4mm \pi\kern8mm$3.14159…
$\kern4mm \rho\kern8mm$air density in kg/m³
$\kern4mm \tau\kern8mm$torque in Nm
$\kern4mm \omega\kern7mm$angular velocity in rad/s

Perhaps the question is trivial

It is not at all, the full explanation can easily fit an entire helicopter aerodynamics book. Let's see if I can fit a proper answer in less space

The speed $V$ seen by a blade is basically given by the sum of 3 different terms:

the speed due to its rotation around the rotor shaft; this speed varies linearly along the bladespan from 0 at the rotor head to $\omega R$ at the tip, where $\omega$ is the rotating speed and $R$ the bladespan;
the speed at which the whole helicopter is flying;
during its rotation, the blade bumps into the wake shed by the previous blade; and after one complete rotation, it bumps into its own wake; these wakes also modify the local speed seen by the blade.

The sum of these 3 terms gives a total speed seen by the blade that continuously changes in space and time! As a consequence we cannot use for a blade the ubiquitous simple lift equation $L=½ \rho V^2 S C_L$ since there's no way to define one single $V$.
But with a workaround we can use that equation anyway: if the hypothesis is done that the blade is composed by a continuous sequence of adjacent airfoils, then that equation can be locally used for each of these airfoils in the sequence.
So, for each slice of blade that equation can be used to get the local lift (and the local drag); afterward the local lift (and drag) is summed up (integrated) along the bladespan, from the rotor root till the tip, to get the total lift and drag; and finally the total lift and drag is decomposed in 1) a force parallel to the rotor shaft, which is the thrust generated by the rotor; and 2) a force perpendicular to it, which gives the torque needed to make the rotor spin. The following picture shows how the aerodynamic force R=L+D generated upon the "slice of blade" can be decomposed as thrust T and torque P:

Obviously that equation can be used if it is know the local speed $V$ on each slice of blade as given by 1. + 2. + 3. As said:

and 2. are know since $\omega$, $R$ and the helicopter speed are known; these two speeds compose $U$;
the speed $V_i$ due to the wake depends itself on the lift and gives rise to a vicious circle: the lift depends on the wake and the wake depends on the lift! This vicious circle is broken by either using simplified models of the wake and/or wind tunnel measurements and/or CFD simulations.

Let's break the vicious circle via a simplified model. If the simplification is done that:

the helicopter is hovering, i.e. term 2. is null;
the wake in/upon the rotor is constant everywhere, i.e. term 3. has a constant value;
the blades have a hyperbolic twist (this is normally true for a propeller blade, helicopter blades have a more linear twist but using a hyperbolic one the math is easier);
and each airfoil is operating at its $\alpha$ of maximum efficiency i.e. maximum $C_l/C_d$;

then we get that:
$C_T=¼ \frac{N_b c_{tip}}{\pi R} C_{l_{\alpha}} \alpha_{@maxC_l/C_d}$
where:

$N_b$ is the number of blades;
$c_{tip}$ is the chord at the tip of the blade;
$C_{l_{\alpha}}$ is the slope of the airfoil's lift coefficient;
and $\alpha_{@maxC_l/C_d}$ is the airfoil's $\alpha$ for maximum efficiency.

Useful references:

Principles of Helicopter Aerodynamics by J. Gordon Leishman
Helicopter Aerodynamics Volume II by Ray Prouty
Aerodynamics of the Helicopter by Alfred Gessow, Garry C. Myers

Thank you for your answer! I would like to see more clearly the relationship between "lift", "thrust" and "torque", these vectors are shown in the pictures below link1 link2

Those pictures are intended to give a simple scheme for pilots to keep in mind. But technically they are simply wrong... A rotor generates thrust (in exchange of torque), full stop. That thrust can be more or less tilted as needed via the cyclic and collective controls to control the helicopter. This is a much more correct picture of what happens.


text:
Can a radio on VOX detect a loud tone and interpret as a cut on transmission (coming from pilot/cockpit)?
Can a radio using VOX detect a short tone and interpret it as a signal for cutoff of audio signal/transmission? this is A/G radio communication using a Schmid radio

This could be a question for Electronics or Ham stacks, but we can state the basics here.

It won't be built-in, neither in the transceiver, nor in the microphone. However such feature can be easily added when the microphone input has a push-to-talk switch (PTT), there is no need to use the VOX feature.
Principle
The principle is to extract the tone from the microphone audio stream with a decoder and control the PTT contact. This is called in-band signaling, and a well-known example is the beep used in Apollo missions to trigger the transmitter of a remote ground station from Houston center. You can insert the device between the microphone output connector and the transceiver microphone socket. The device controls the PTT pins of the transceiver input.
Technology
There are really many solutions of different types. You can use discrete components (transistors), an integrated circuit like a phase-locked loop (PLL), an general purpose operational amplifier (µ741), or a numerical solution (digital signal processing).
A cheap LM567 PLL decoder ($2.81 at Farnell) seems a good choice. The frequency to detect is adjusted with a resistor and a capacitor (the decoder work up to 500kHz). If you use two simultaneous tones to increase reliability, two decoders can be used, but it would be better to use the DTMF standard and a ready-to-use decoder.
Schematics
Schematics for tone detection with a LM567 are available online, e.g. this one from Codrey Electronics:

Audio is provided on 3, the output is pin 8. You need to replace the LED by a Reed relay or an optocoupler, to control the PTT pins. You can get more guidance on the stacks I mentioned at the beginning.

There's no reason why this couldn't be done, but I am not aware of any system implementing such a technology. One potential problem is that something else in the cockpit could make such a sound and cut the transmission; alternatively, background noise could drown out the tone. Perhaps a better option would be to simply have a button to stop the transmission; some kind of device to generate the tone would be needed anyway, so it makes more sense to have whatever input would have gone to the tone generator go directly to the radio.

An update regarding the topic. This question originated from a coupling issue. Two frequencies are coupled, and the retransmission frequency was found to be the source of the tone.

@CommSys_noob01 oh, sorry; I misunderstood your question. I thought you were asking if having a tone cut the transmission could be implemented on purpose as a feature. Did you get your problem fixed?

Nope, not really. there's still this tone present but it's thought to be the PTT of the other radio(the retransmission freq) which is coupled.

@CommSys_noob01 you need to edit that info into your original question, not bury it down here in the comments on an answer.


text:
How can a plane or an object gimbal lock itself?
I've been trying to work out some rudimentary dead-reckonings with an IMU chip, and most of the kinetics/kinematics and algorithms are borrowed from aviation related textbooks.  I understand what the Euler angles are, how the rotation matrix originates from a dot product between frames, or how to convert my Euler angles to Quaternion to run the algorithm etc.  But I just couldn't understand gimbal lock at all.
There are plenty of resources on the web (e.g. Matthew Brett's Gimbal Lock, J.Gallifent's Gimbal Lock, and most notably this YouTube video Euler (gimbal lock) Explained) and a good number of related if not identical questions on SE (most related if not practically identical questions Euler Angles - Gimbal lock, why non-orthogonal axes Gimbal lock confusion and Euler-angles-and-gimbal-lock).  Even If I can follow the math derivation, I could not make any qualitative sense or visualization of the concept if the object is not mounted on a gimbal or rings (why would I ever want to mount a plane or an IMU on a gimbal anyway).

The basis of any frame $(\mathbf{i},\mathbf{j},\mathbf{k})$ is always orthogonal, and each of the three rotation matrices in $R_x(\phi)_2^3R_y(\theta)_1^2R_z(\psi)_0^1$ transform the basis all together, from one frame to another.  In between or throughout the three rotation transformations, the basis stays orthogonal - the nose always points in the $+\mathbf{i}$ direction, the right wing to $+\mathbf{j}$ and belly of the plane faces the $+\mathbf{k}$ direction.  No order of rotation and not one value of $\psi, \theta, \text{or } \phi$ can possibly align $\mathbf{i}$ to $\mathbf{j}$, or $\mathbf{j}$ to $\mathbf{k}$.
The only ways to align two axis to satisfy the condition needed for gimbal lock is to mechanically deform the object, such as bending the nose $90^{\circ}$ around the $y$-axis to align $\mathbf{i}$ and $\mathbf{k}$, or chopping off both wings and attaching them to the nose to align $\mathbf{i}$ and $\mathbf{j}$. But that's just non-sense.
I think I'm in need of a qualitative answer or an example of a plane gimbal locking itself to understand what gimbal lock is.  Or correct me if I have an entirely erroneous concept of gimbal lock.

[EDIT]
Adding a quote from my textbook Small Unmanned Aircraft Theory and Practice by R. Beard (p.15-16):

The rotation sequence ψ-θ-φ is commonly used for aircraft and is just
one of several Euler angle systems in use. The physical interpretation
of Euler angles is clear and this contributes to their widespread use.
Euler angle representations, however, have a mathematical singularity
that can cause computational instabilities. For the ψ-θ-φ Euler
angle sequence, there is a singularity when the pitch angle θ is ±90
deg, in which case the yaw angle is not defined. This singularity is
commonly referred to as gimbal lock.

You didn’t offer a quote from whatever text mislead you, but from a series of comments it appears that your question is based on a simple misunderstanding.
Aircraft and other objects cannot gimbal lock.  The only thing that is capable of being gimbal locked is a gimbal itself.
Aircraft use gimbal mounted gyros for inertial navigation and attitude reference.  The gimbal is required to allow the gyro to remain level relative to surface of the earth.  These gimbals may lock up during certain aggressive maneuvers.

Appreciated. I have yet to fully comprehend some of the details but at least I think I know now what my problem is.  Coming from a background with no practical knowledge (or common sense) in aviation, I see an IMU or navigation unit in general as a chip mounted on a PCB.   How could then a piece of circuit board behaves like rings on a gimbal? .... until I looked into some of the videos and saw what's inside an attitude meter - there are real spinning gyros and gimbal like frame!

@KMC, You are most welcome!  Lots of planes do still have spinning mechanical gyros.  It's pretty cool and they still work.  Glad I could help, and I hope my poking at you a bit was taken in the spirit it was intended.  (I removed my DV by the way...)

When you measure angle in Euler angles (yaw/pitch/roll a.k.a. pan/tilt/roll) you are imagining that the object is mounted in these gimbals and then you are measuring the angles of the imaginary gimbals.
If the imaginary gimbals line up in a gimbal lock way, it can cause a high rate of change to the Euler angles which can cause numerical errors akin to the ones caused by a real gimbal not being able to move quickly enough.
The extreme example is when the pitch and roll axes line up exactly. Imagine that you are looking straight forward, then you tilt your head back and look straight up, then you keep tilting it in the same direction (breaking your neck) and you look straight backwards. When you pass through the straight up direction, your yaw and roll instantly flip 180°. Computations involving derivatives of yaw or roll may throw a fit. If you pass nearby that direction but not straight through it, you will not see an instant flip but a very rapid change. If you are looking straight up, small deviations, e.g. caused by sensor inaccuracy, can result in very large numerical changes to your measured angle.

Everthing make sense ONLY if I, as you said, imagine a plane is mounted on a gimbal . But why do I want to imagine there to be a gimbal in the first place? The plane has no rings. Its nose(roll), belly(yaw) and wings(pitch) stays orthogonal to each other all the time and so is each individual rotation caused by the Euler angle

@KMC Because that's what Euler angles are. Euler angles are imaginary gimbals. If you use rotation matrices or quaternions for processing, then you don't imagine gimbals and you don't have gimbal lock.

In this case, I guess I'm not understanding how the Euler angles relate to a gimbal.  Mathematically I think I do if I plug in a 90deg to one angle and zero out a term, but not qualitatively or the intuition.  Euler angles suppose to rotate the entire frame with its basis which stays orthogonal throughout. It doesn't rotate like a gimbals where all three rings doesn't necessarily rotate together.  So I still don't see how Euler angle mechanically relate to a gimbal. They are just two different things.

When a inner gimbal ring rotates, the outer ring doesn't. But when the first Euler rotation rotates, all three axis rotate...

@KMC because the first Euler rotation is the outer ring

Even so, regardless of the order, when the first angle rotates, i1,j1,k1 rotates. When the second angles rotates, they all rotates and remain orthogonal to each other - i2,j2,k2.  Whatever is applied as the last rotation, the frame lands on i3,j3,k3.  How is Euler rotation has anything to do with gimbal?

This is all way above my competence, but from what I gather, it seem lika a problem regarding the frame of reference and coordinate system(s). An aircraft's axis are not interlinked in any way. A gimball does not have this absolute freedom.The rings are sequentially interlinked.

@Jpe61 there is something fundamental I'm badly missing here.  An aircraft's axis IS  perfectly interlinked (its nose, belly and wings are permanently orthogonal - if that's what you mean).  A gimbal is sequentially linked and is a structure by itself that draws no relation to a physical plane.

@KMC they are interlinked in physical sense (sorry for layman terms), but as a simplified model, any movement around any axis will not ever limit movement around any other axis. IRL this is of course different beacuse of aerodynamics, gravity etc., but again, in comparison to a gimball, there is no condition that will link movements around any two or three axis in a manner that would lock them together preventing free movement. I'm afraid I'm not able to communicate what I see as self evident by intuition here.

I'm beginning to see my problem. There are actually physical objects that behaves like a gimbal installed in a plane although I'm not sure why are there gimbals or why they don't just glue/fix the rings orthogonally so the gimbal will never align their axis.  I should be reading why people put gimbals in an aircraft.  I though it was a math phenomenon that happens to be explained with a gimbal.

So gimbal lock is about real gimbals people put inside an aircraft when the plane is rotated by Euler angles....

@KMC There don't have to be real gimbals. Imaginary gimbals can lock just like real gimbals. Euler angles are imaginary gimbals. If you use Euler angles then you are using imaginary gimbals. There's no way to use Euler angles without using imaginary gimbals. Imaginary gimbals can cause data processing errors in a way that is extremely similar to how real gimbals can lock, so we can call it gimbal lock. Nothing happens to the aircraft. Only the computers mess up. (If the computers are controlling the aircraft and they mess up, maybe the aircraft will mess up!)

@user253751, Euler angles couldn't possibly be imaginary gimbals, at least intrinsically.   Each rotation in the sequence of Euler rotation maintain the orthogonality of all three axis (if x-y-z starts out to be perpendicular, they will always be perpendicular to each other regardless of the chose of sequence of rotation).  It is the Euler rotation we applied to an object $\emph{confined}$ in a gimbal that causes the object to appear being locked because two rings aligned.  That's my understanding so far.

@KMC Real gimbals also maintain the orthogonality of all three x-y-z axes of the object inside the innermost gimbal

Your aircraft will only gimbal lock if you treat it as if it were a gimbal. That is, you would consecutively rotate it around each one of its three axes until it has been rotated around all three.
Numerical gimbal lock means that you first rotate around one axis, then around another, and by happenstance this is a rotation around 90° which aligns the third axis with the direction of the first (in the global axis system that does not rotate) such that the third rotation will only allow a movement which has already been performed by the first rotation (again, in the global reference system). This removes the freedom to rotate around all three axes freely, because the first and the last rotation could be summed up in a single rotation. This is equivalent to a reduction in the degrees of freedom by one, and just the same as with a gimbal of which two axes of rotation align.
I hope you will agree that aircraft do not rotate in this way in real life.

To your last sentence, I agree that aircraft do not rotate this way in real life, but you are implying that if you were able to perform a series of aerobatic maneuvers in the way you describe that the aircraft itself would gimbal lock.  Is that the intent of your answer?

@MichaelHall No, not really. You cannot restrict rotation of a real thing to a single axis at a time the way you do when doing an Euler angle transformation. That is my point.

But you are saying that an aircraft "will" gimbal lock if you were to treat it as if it's a gimbal.  How is this possible?

@MichaelHall If you fix it into rings with bearings (huge rings, admittedly) like a gimbal, which when their axes are lined up will restrict its degrees of freedom, then yes, it will gimbal lock.

Ok, maybe this is a matter of semantics, but it seems to me that only the gimbal itself can lock, and not the object in the gimbal. I get your point, it just seems this answer is playing into the misunderstanding at the root of the question. (that some object other than a gimbal can lock up...)

The phrase "gimbal lock," as your textbook uses it, doesn't really have anything to do with gimbals (except metaphorically). Your textbook uses the phrase "gimbal lock" to refer to a situation where small changes in attitude result in large changes in Euler angles.
Suppose that an airplane is flying due north, straight and level. The heading is 0°, the pitch is 0°, and the bank is 0°. Then the airplane pitches up by 89.9 degrees. At this point, nothing weird happens: the heading is still 0°, the pitch is 89.9°, and the bank is still 0°.
Now suppose that the airplane pitches up by another 0.2 degrees. The Euler angles change dramatically! The nose of the airplane is now pointing south rather than north, and the airplane's belly, rather than its roof, is above the horizon. As a result, the heading is now 180°, the pitch is still 89.9°, and the bank is now 180°. A mere 0.2 degree change in attitude caused both the heading and the bank to change by 180 degrees!
That's the phenomenon which your textbook calls "gimbal lock." There aren't actually any gimbals, and nothing is actually locking, but weird things are happening with the numbers. The reason that this is called "gimbal lock" is that the mathematical formulas that describe what's happening here are the exact same mathematical formulas that describe real gimbal lock.
As you would imagine, gimbal lock in this sense doesn't actually affect the aircraft itself in any way whatsoever. It's an artifact of the way that we do calculations related to aircraft; it's not an event that actually physically happens.
Solid-state and MEMS gyroscopes aren't physically affected by gimbal lock either. (Your IMU chip probably contains a MEMS gyroscope.) But if the software on them is incorrectly written, then it may malfunction.
(By the way, you might have noticed that we could avoid the dramatic "flip" above by saying that the heading is still 0°, the pitch is now 90.1°, and the bank is still 0°. But now suppose that, starting from north-straight-and-level, the airplane pitches up 89.9 degrees, then it pitches up 0.1 degrees and yaws right 0.1 degrees. Now the heading is 90°, the pitch is 89.9°, and the bank is 90°. There's no way to avoid the sudden 90 degree change in heading and bank.)

Onto the last paragraph, if the North-heading airplane pitches twice $89.9^{\circ} + 0.1^{\circ} $ the nose (or the x-axis) points straight up and after the last $0.1^{\circ}$ yaw, the plane's belly faces due North and its nose East.  Either the heading or the pitch can be $90^{\circ}$ and the bank would be $0^{\circ}$.  Why is the pitch $89.9^{\circ}$ and the bank $90^{\circ}$?

@KMC Pitch is the angle between the nose and the horizon. If the plane had yawed 90°, the nose would be on the horizon, giving a pitch of 0%; but since the plane yawed by only 0.1%, the pitch is 89.9°. The only way to have a pitch of 90° is to have the nose pointing exactly straight up.

@KMC Bank angle is the amount that the aircraft would have to roll in order to bring the wings level with the belly facing down rather than up. After the plane pitches up 90° and then yaws right 0.1°, the wings are no longer level: the left wing is 0.1° above the horizon and the right wing is 0.1° below the horizon. In order for the plane to bring its wings level again by rolling, it would have to roll 90°.

@ Tanner - reinstate LGBT people, aha! This answer along with the comments is gold! I guess why weird things can happen then has to do with how the instrumentations are mechanically design or if they are sensed and processed numerically the fixed X->Y->Z ordered rotations can interpret one small change as big rotation around the other axis. Hence we need to apply a different set of Euler order or move on to quaternions.  I was having a hard time disassociating a plane from a gimbal, then to relate the math to what is physically measured, but I think I'm getting the picture now.


text:
How is rotational dynamics related to navigation?
I've been learning how to do dead-reckoning using just an ordinary IMU chip (a 3-axis accelerometer and a 3-axis gyroscope). In a number of textbooks I've read on aircraft navigation, the motion of a navigating object is often summed up in four equations.  Aside from the last equation - (3.17) rotational dynamics - where we need to separately calculate the inertial matrix in CAD, the other three equations requires only the input data registered by the accelerometers or gyroscopes, or integration of the values.
I have so far no problem understanding the derivation for each equation or knowing what they physically mean. The trouble I had is I couldn't relate the rotational dynamics equation (3.17) of an object to its navigation.
Navigation is a matter of tracking the position and orientation of an object and perhaps also its acceleration and velocity at a specific point in space. The first three equations (3.14) (3.15) (3.16) will suffice:

Given an initial condition and gyro data, we can numerically update the Euler angles in (3.16) and determine the object's orientation
With data measured by accelerometers and gyros in the body-frame, and if we know our externally applied force, we can find out the object's velocity in (3.15)
Since we already know our orientation and body's velocity, we can calculate our inertial velocity in (3.14). Integrating that, we get the object's inertial position

Up to this point I would have everything I need for navigation and the last equation (3.17) seems out of place here. I've already had my orientation per unit time - what's the point of knowing the angular acceleration around the bodily axis? Calculating all the different moments of an object doesn't seem to have any meaning to navigation of an object either. So why does that equation (3.17) matter in navigation?

[Edit]
I start to suspect it could be my entirely lack of basic knowledge in aeronautics or aviations that gives rise to this question.  I don't see any mathematical difference in mounting an IMU on a banana or on an actual aircraft. The IMU provides me with accelerations and angular rates that I can calculate where the object navigates in what orientation. Navigation care less for the inertia or moments of the object. Even if I need angular acceleration for whatever reason, wouldn't it be much easier (if not more accurate) to differentiate the gyros data directly?
The only situation I can think of that makes inertia and rotational acceleration matter in navigation is when the object collides into something else where a transfer of momentum occurs. But that's just my wild imagination and am unsure if that makes any sense.

Will your IMU be affected by body rates?

I’m a little rusty here but generally in spherical motions, you have to account for the Euler term AND the Coriolis term.  Except at the equator, a great circle line always has a northern or southern component, irrespective of the direction of motion.  That northern or southern component, if performed at a constant altitude, mean I am moving inward or outward from the vertical axis of my motion, and I am going to have to account for that in my navigation solution.  I believe that that’s 3.17, it’s where you account for the conservation of angular momentum.

What do p/q/r represent in the equations? looks like something related to angular velocity? Thanks to the Dzhanibekhov effect an object with constant angular momentum might not have constant angular velocity though I don't know if that's what this equation means.

@Jim If you're referring to spinning of an object, then no.  I only care for the orientation and position (and it's derivatives) of the IMU.  If by body rates you mean linear and angular velocity ... well that's what the IMU is measuring.  Otherwise I'm not sure if I understand your question.

@Max R, I believe the equations are in their most general forms. Acceleration $\mathbf{a}(t) = \left(\ddot{r}-r\dot{\theta}^2\right)\hat{\mathbf{e}}r  + \left(r\ddot{\theta}+2\dot{r}\dot{\theta}\right)\hat{\mathbf{e}}{\theta}$ encapsulates both radial and tangential terms (Coriolis acceleration being the last term). But this fraction of acceleration shouldn't have anything to do with navigation. It really is just a secondary abstraction to be calculated from the data in hand (e.g. find out the Coriolis force etc.).  After all, each numerical data is as linear as it can be per unit time.

@user253751 $(p,q,r)$ are angular velocities measured by the gyros and $(u,v,w)$ are linear velocities or integration of accelerations measured by accelerometers. Both sets of data are measured in the body frame.  I'm also not familiar with these aeronautical terms and prefer basic physics symbols of $\omega^b_x$ instead of $p$.  But it seems they are standard terminologies in navigation and we're stuck with it.

Accelerations derived from differentiating velocities are 1 or multiple frame rates behind the real values, which could be a stability issue in numerical computation.  And for the banana - you need to know which way it is pointing, right?

@Koyovis The question is purely conceptional. I assume all data and numerical integration are ideal. In practice we'll be computing quaternions with different filtering and integration algorithms. Here, I ignore any singularity and noise to focus on fundamental understanding.  As for the "banana", of course we want to know where it's pointing at and (3.16) already gives us the change in orientation and if given an initial orientation (easily with accelerometers measuring the gravitational vector) I can get the orientation at any given time. So there's no need for (3.17), and hence my question.

I'm not 100% familiar with the symbols .. anyway. For my understanding.. am I right that 3.16 is  only a transformation from IMU reference frame to body RF - if they were aligned, it should be unity matrix? Coriolis and Euler Terms I see not only valid for spherical navigation but also if the IMU isn't located in the center of rotation (~CG): the IMU will then see linear accelerations which are only aircraft rotations & rotational accelerations "multiplied by the radius CG-IMU". Differentiating is super noisy, you're likely  better off transforming accelerometer measurements.

@CarlBerger, to your first question on (3.16) - yes.  Regarding noise, differentiating data from gyros are not that bad, and also fortunately I've only have to deal with integration for the cheap and super noisy accelerometers.  But here I'll treat all data to be perfectly true and ideal just for the sake of the question.  That is, why the heck do I need to know about angular momentum and moment of inertia in navigation?  Equation (3.17) doesn't seem to contribute any improvement to the calculation or prediction of position and orientation, in my poor common sense in this subject that is....

As you noticed, you do not need this information for navigation per se, and you are right. If you can measure the rotational rates $p$, $q$ and $r$ directly, then you do not need to worry about updating the rotational rates.
However there are a couple of reasons why you might be interested in this equation anyways.

This completes the set of equations. These are generally called the equations of motions of an aircraft and are central for flight dynamics. Therefore this set of equations describes how an aircraft moves/rotates/behaves in the air.

For real-world navigation you also might use this formula for example inside a data-fusion scheme. No sensor is perfect, therefore directly using the rotational rates for your navigation solution might lead to undesirable results. For this reason, often data fusion approaches like a Kalman Filter are used which fuse sensor data and model data. For the model data part of this filter you need a mathematical model, which is exactly posed by the rotational dynamics. (Probably related: this question)

Last but not least, these equations are used for simulation purposes. If you want to numerically simulate an aircraft for whatever purpose, you absolutely need these equations. Simulations are of course central for a whole myrade of applications such as flight simulators (e.g for testing your navigation filter), general flight dynamics analysis, controller synthesis, and so on.

From the context you provide, I would speculate that the author wanted to give the complete set of equations.
Perhaps as a side-note regarding this set of equations: These are flat earth approximations (not related to the conspiracy theory :D). Therefore not really suited for navigation purposes if you want to navigate further then lets say 30-50km. After this distance, the error accumulates and gives distance errors in the 10s of meters. In applications like short term path planing like landing planing or short-term path following, these are fine however.


text:
What is the relation between body-frame, body-fixed-frame and vehicle-frame?
When I first came across these three terms, I thought they were all the same. A frame attached to the body of a plane and centered at its center-of-mass with the frame's $\left\{\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}\right\}$ aligns with the plane's { nose, wings, belly } respectively.  If the plane starts out heading north and yaws $90^{\circ}$, the new $\hat{\mathbf{i}} $ will then point East. Since the body-frame is fixed to the body of the plane, we can also called it body-fixed-frame. The body of Micro Air Vehicle (MAV) is termed as a vehicle instead of a plane, and thus the name vehicle-frame.
$$ \text{body-frame} \equiv \textrm{body-fixed-frame} \equiv \textrm{vehicle frame} $$
I have always treated these terms as context-dependent synonyms until I see this following equation in my textbook (Small Unmanned Aircraft, Beard) where it rotates the body-frame to vehicle-frame! What? "Body" and "vehicle" are different?

I couldn't make sense out of it. Since this equation is formulated on translational motion, I can treat the plane (or the vehicle) as a particle positioned at its center-of-mass (or origin of the body-frame).  Any instantaneous translational velocity experienced by this particle interpreted by it's own body-frame is $ (\mathbf{v}^b_x,\mathbf{v}^b_y,\mathbf{v}^b_z)\equiv (u,v,w)$.  If we want to interpret these velocities under the inertial reference frame $(\dot{p}_n,\dot{p}_e,\dot{p}_d)$，shouldn't we just rotate the body-frame to the inertial-frame instead?
$$ \frac{d}{dt} \; \left(
  \begin{array}{ c }
     p_n \\ p_e \\ p_d
  \end{array}
\right) = \mathcal{R}^i_b \left(
  \begin{array}{ c }
     u \\ v \\ w
  \end{array}
\right)$$
What does it mean to rotate a body-frame to a vehicle-frame ... isn't a body already a vehicle? And if rotating a body-frame to a vehicle-frame gives us values projected onto the inertial North-East-Down axis as the textbook suggest, wouldn't that makes the vehicle-frame aligned with the inertial-frame?
I've tried googling but with no success, and these terms are also not clearly defined in the textbook.

Just to get this right, you are asking in reference to the transformation $\mathcal{R}^v_b$ depicted in your picture, right? Because the text of the textbook does not mention vehicle frame...I would say that this is a typo. It must be meant $\mathcal{R}^i_b$ because then the equation from your textbook is quite familiar...

@U_flow yep get it right - why the book goes with $\mathcal{R}^v_b$ instead of $\mathcal{R}^i_b$.

I believe what's happening with the nomenclature is this:
$\mathcal{F}^i$ is the earth NED frame
$\mathcal{F}^v$ is a NED frame centered at the vehicle CG
$\mathcal{F}^b$ is a uvw frame centered at the vehicle CG
$\mathcal{F}^{v1}$ is the frame after the first Euler rotation ($\psi$)
$\mathcal{F}^{v2}$ is the frame after the second Euler rotation ($\theta$)
$\mathcal{F}^{v3}$ is the frame after the third Euler rotation ($\phi$), ie. $\mathcal{F}^{v3} \equiv \mathcal{F}^b$
So vehicle frame isn't just a MAV-centric term for body frame- it is a NED oriented frame attached to the body.


text:
What odd maneuver is this for a cruising airplane?
Several years ago, I saw a set 4-5 of con-trails converging at nearly the same time and height. They were each executing large turns (90 to 300 degrees) over the same space, starting from various directions, and leaving from various. This took over/near Kansas City Int'l (MCI).
Sadly, I failed to take a pic.
Recently, I got another pic! Same location, apparent altitude, same high-degree turn, but only one con-trail.
What would cause a plane (or 4 planes!) to make such large turns? Surely diverting to a new airport would only cause a minor deviation from cruising alt.
Both times I was 17.5 mi south of MCI.  I think I've probability mis-characterised their altitude.

Once in a while I also see something very very similar to that over my head and they are from jet fighters... training or just having fun :)

Can you give us an idea of where this is? There may an area they are not allowed to overfly, and they have to make this little "detour". Or it may be the junction between two airways which avoid that area. For instance North of Las Vegas there's a large area with no airways, so the airways on the sides have pretty large angles between them.

Sorry, I missed that you wrote this was near MCI. Can you let us know the direction you were facing when you took the picture, and possibly where relative to MCI you were? Also, did you check on flightradar24 or flightaware for any flights around that date/time?

Both times I saw this, I was 17.5 mi due S of MCI.  I.e., facing north.  I'm probably wrong about them being at cruising alt.

Is the question about the single "bright" trail that turns, or the multiple wispy faint trails that diverge progressively from the line that looks like the edge of a thin high cirrus cloud? The latter are more interesting to me! The shape looks like a military "overhead break" where a formation splits to land, but that's not normally done at high altitude.

If you look at the airways in the area on skyvector.com you'll see it could be an aircraft coming from J80 or J26 going to J41 for instance. Depending on where they are coming from/going to it's probably not an ideal routing, but there could be some issue on the more direct routing that would have sent them this way. Next time you see this, look up flightradar24 (you can install the app on your phone) to see what flight it is, it could help explain exactly why they are doing this. Note that it is quite possible it happens very often but you only notice it when they leave contrails.

The question is about the single, bright con-trail.

They were probably "stacked up" in a holding pattern while waiting for permission to land at a busy airport. This is a common occurrence and the flying maps that pilots use will have a "racetrack" marked up near the destination airport. If things are too busy on the ground to manage incoming traffic, the air traffic controller will instruct the pilot of an arriving plane to divert and "hold as published", which means proceed to the racetrack and go round in circles until you get the all-clear instruction.
Each plane in the holding pattern is assigned their own altitude to circle in, and the ATC will pluck the bottom-most plane out of the stack and clear them to land. Then all the other planes that are stacked up are reassigned the next-lower altitude to hold on and the whole stack "drops" down one level.
This puts the plane that has been holding the longest at the bottom of the stack, and the newest plane to arrive is inserted into the topmost level in the stack.
ADDED IN EDIT:
As pointed out by others here, course changes like the one pictured are often performed by military aircraft, performing maneuvers on training flights. You can pick these out from the ground by noting that such flights do not generally follow the airway patterns for your area. For example, in western Oregon between Eugene and Portland, all the commercial flights high enough to pull contrails follow north-south lines. A jet at cruise that flies overhead here on an east-west track is therefore not a commercial flight because there are no commercial destinations along that line.
BTW once on a flight between Portland and the SFO bay area, our pilot announced that ATC wanted us to arrive 5 minutes or so later than scheduled and in response the pilot made a slow right circle, at cruise. This was approximately over the Oregon-California border. I do not know how often passenger jets get instructed to "hold at cruise" like this.

I've heard of racetracks, but I've never heard of them being at cruising altitude before.  Can that be right?  Do planes really circle up to 200 miles from the airport?

yikes.  If the 5000' plane doesn't get word to descend to 4000', then the 6000' plane will smack into it when it descends to 5000'.  You'd need a positive means to ensure the 5000' guy got the order.  Obviously you can do that once or twice safely enough... the problem is, you have to get it right every single time.

@Harper, This is routinely done every day. In an ATC's training, they are required to drop a stack of five planes or more sequentially, by memory, in case their radar displays fail and they have to manage the racetrack by hand.

@Harper-ReinstateMonica You can be sure that it's a solved problem. Basically any flaw in the system, that you can think of, if you aren't heavily involved in the matter, has been dealt with decades ago.

This answer is accurate in describing holding, but I’m not convinced it’s applicable to the photo.  All I see is a course change.

@MichaelHall I'm not convinced either, although I suppose it could be a direct entry to a holding pattern, assuming the aircraft is flying away from the camera and then turning clockwise.

@Michael - Planes can't circle 200 miles "straight up" - that would be into orbit!  Cruising altitude is ~32k feet, only 6 miles, but the stack starts far below that.  From a quick back-of-the-envelope calculation, it appears that a leg of the track is ~4 (nautical) miles, so they're not going 200 miles from the airport horizontally either...

I can’t imagine a stack ever forming at MCI. It’s just not that busy.

I question the idea of planes stacked in a holding pattern up at the contrail level, regardless of specific location.

@Clockwork-Muse -- That was the point of Michael's comment too-- regardless of the validity of the specific "200 mile" figure, if you are holding that high (high enough to leave a contrail), you still have a long linear descent leg ahead of you -- unless you are following a really inefficient flight path. (No one was suggesting that planes were circling at an altitude of 200 miles, or that the actual holding pattern included a 200-mile leg!) –

In my experience as an observer, aircraft performing large turns (greater than say a simple 45-degree "bend" in the flight path) up at the contrail level have a greater-than-average chance of being military rather than commercial aircraft.  Always a good reason to break out the binoculars and see what aircraft is leaving the trail.
From your description, it's difficult to tell whether any of the aircraft were actually flying together in close proximity, or not.  If so, aerial refueling comes to mind as a possible explanation.  But one can imagine many other scenarios, including various training exercises, where one or more military aircraft would follow a flight path that was configured to prioritize objectives other than simply transiting from point A to point B as efficiently as possible.

The usual objection about not invalidating existing answers notwithstanding, perhaps the question could be modified to address the last paragraph of this answer --

Both times I was 17.5 mi south of MCI.  I think I've probability mis-characterised their altitude.  No known military activity that I know of there.

A tanker on an aerial refueling mission seems like the most probable explanation... especially for the not-photographed case of several contrails doing similar things... one or two tankers & multiple receivers, all following similar tracks.

Edited question to remove mention of plan(es) being at cruising altitude. I do not the altitude.

@user1745937 -- to me the edit doesn't really change anything, and wouldn't have affected my answer, the point is that they were high enough to make contrails, which is pretty much only going to happen up at the levels that a jet would consider using for long-range cruising flight, or at least somewhere kind of close to that.   (Google "at what altitude do contrails form").  And to the best of my knowledge, the "holds" mentioned in other answers are not routinely executed that high, though one answer does detail one specific exception, a single circle performed to delay the arrival.

@nielsnielson -- PS one reason that we know that it's rare for traffic to be holding up at the contrail levels, is that if this were common, the chem-trail conspiracy buffs (see https://en.wikipedia.org/wiki/Chemtrail_conspiracy_theory ) would be constantly talking about aircraft spraying chemtrails while flying in ovals.  But they don't.  Instead, they talk about spraying while "flying in a grid pattern"--

-- which is actually exactly the expected pattern resulting when the whole airmass is advecting (moving) diagonally, in relation to a couple of well-used linear airliner "tracks" that happen to intersect at a near 90-degree angle.


text:
Is my equation for calculating ducted propeller thrust correct?
The equation is 13(EfficiencyxProp diameterxoutlet diameter)^2/3
It seems really high when I use inches and really low when I use feet.

Please edit your question so it is clear what equation you're actually using - as is, it's not clear. Also, do you have a source for this equation? A properly phrased equation will note what units are used for each term, both input and output... after all, an equation that yields thrust in ounces will have at least one different constant than an equation that gives thrust in pounds.

This is a quick tutorial on how to write equations on Stackexchange

Ralph J- No I don’t have a source. Thanks for the tutorial sophit

Is my equation for calculating ducted propeller thrust correct?

Almost
It should be something like:
$T=1.3 \eta P^⅔$
but this is valid only on one specific case.

The easiest equation relating thrust $T$ and power $P$ is the one derived from the simple momentum theory:
$T=\sqrt[3]{2\rho A P²}$
where $\rho$ is air density and $A$ is disk area i.e. $πr²$. In this equation you have:

the "$^⅔$" applied to $P$, just like in your equation;
the propeller's diameter $d$ (actually its radius $r$), just like in your equation;
to take into account the duct, the thrust is normally increased of some 30%, the power being the same; and here you get also the 1.3 (not 13) just like in your equation;
and to take into account all the effects which are simply disregarded in the momentum theory, the right part can be multiplied by an efficiency factor $\eta$ (some 0.6 to 0.8), just like in your equation.

Rearranging the terms and introducing the factors just discussed, you get:
$T=1.3 \eta (Pd)^⅔\sqrt[3]{½\rho π}$
which resembles what you were actually looking for. And if you consider the density at 3km (10'000ft) height and a propeller with 1m diameter, then the equation becomes exactly what you remembered:
$T=1.3 \eta P^⅔$
Note that all these equations are valid only at rest (zero speed).

It seems really high when I use inches and really low when I use feet.

Please use SI units.

This is really helpful and makes a lot of sense. My memory may have been faulty, which explains the 13/1.3. S.I units are metric right? Like meters for length?

@Lelotz: correct Watts, meters, kg, Newtons, ...

The short answer is "No" it is not correct.  Even though the equation formatting is lost we can still know that it cannot be correct because the quantities shown are efficiency, and two diameters.
Efficiency is dimensionless and the diameters have units of length whereas thrust as units of force.  length cannot be combined with itself in any way to get force.  Of course there is another number in the equation- a 13.  And that could be presumed to have the units required to get force.  However, if that were the case it would mean that the resulting thrust would be independent of anything outside the equation.  For example engine horsepower.  And we intuitively know that for a given efficiency, prop diameter and outlet diameter the thrust can be changed dramatically by using different amounts of engine power.


text:
Is the Garmin G1000's heading indicator based on a Magnetic compass?
How does the heading indicator displayed on the PFD of the G1000 work?
A conventional cockpit uses a magnetic compass to correct the heading indicator.
However, I don't know if the heading indicator on the monitor of the G1000 detects and measures the magnetism itself and displays it on the monitor, or if there is another principle.
Of course, I know that the GMU-44 LRU is equipped, but I wonder if it is measured and output as PFD or if there is another principle.

The Horizontal Situation Indicator (HSI) rosette display on a G1000 works in a similar fashion to a conventional HSI operating in slave mode.  It is slaved to a 3 axis magnetometer which gives it heading cues. A magnetometer makes use of a flux gate or flux valve consisting of a 3 part iron ring fitted with 3 pickup coils on the arms and a single excitation coil around the core.

The excitation coil is energized with an alternating current.  When this AC current reverses direction, it allows the pickup coils to sense the magnetic flux of the earth’s magnetic field.  How intense the flux through each of the pickup coils is is determined by how the ring is aligned with the Earth’s magnetic field. See image below.

Therefore, the net electrical inputs from these pickup coils can be interpreted via either an analog or digital computer and related to the direction of the aircraft relative to the Earth’s magnetic field.
Flux gates are typically mounted in the wing of an aircraft near the tip well away from avionics or other electrical systems to minimize risk of magnetic interference and consequently heading errors from magnetic deviation.  When installed and calibrated correctly, the magnetometer should indicate a heading either on or very, very close to what is indicated on a free moving magnetic compass.  This should be verified during preflight and taxi checks that these two navigation devices operate in harmony.

TL;DR: yes, it's a magnetic compass.

Negative. It is not a magnetic compass. The equipment and functioning are different between the two items.

But you've just said that it reads the surrounding magnetic field.  That makes it a magnetic compass (an electronic one, rather than mechanical, of course).  It's clearly not a gyro.

Not exactly. That’s not how a magnetic compass works either.

This is an awesome answer. And no, this is not a magnetic compass. It uses a magnetic field as a signal source, but not in the same way and requires additional processing, alignment, and calibration.

Okay, I'm clearly missing something, so I've extracted the disagreement into a question of its own.

A good way to figure this out is to look up the annunciation modes of the Garmin display. Here's a Cessna-specific version of the G1000 that I grabbed.

This page shows the red X will happen from the GMU44 magnetometer or from the GRS77 AHRS. The latter uses a gyro (basically). Both the AHRS and magnetometer are used to initialize the heading.
As seen in the appendix, the '77 uses GPS for a backup operation. So there are failure modes for magnetometer, gyro, and GPS.

One reason they use the AHRS like you've indicated is to prevent short-term errors, whether from nearby ferrous anomalies like the runway at London City Airport, or from trying to keep up with a rapidly turning aircraft.


text:
Is my airspeed calculation code correct?
Not sure if I should ask here, on Drones SE, or on Code Review SE.
Is my Python code for calculating airspeed given ground speed, ground direction, wind speed, and direction from which the wind is blowing correct? It takes directions in degrees, and the speeds can be in any units as long as they're the same; the output will be in the same unit.
import math

def find_airspeed(ground_speed, ground_direction, wind_speed, wind_direction):
    angle = math.radians(wind_direction - ground_direction)

    return math.sqrt(
        (ground_speed + math.cos(angle) * wind_speed) ** 2
        + (math.sin(angle) * wind_speed) ** 2
    )

Or, expressed as formulae:
$$\theta = (\Theta_{Wind} - \Theta_{Gnd})$$
$$AirSpeed = \sqrt{(V_{Gnd} + V_{Wind}\cdot cos(\theta)) ^2 + (V_{Wind} \cdot sin(\theta) )^2}$$

This kind of question should be broken into two parts.  First check the derivation of the equation itself.  I.e., the physics and math portion. Once you are sure the math is right, then focus on whether the code faithfully reproduces the calculation.  We shouldn't have to derive the equation ourselves just to check your work.  :-)

Do not forget that, since you are summing up flight speed and wind speed, then wind is positive when investing the aircraft from nose to tail.

Maybe this can help: https://aviation.stackexchange.com/questions/46741/is-this-the-correct-sequence-of-steps-to-convert-from-true-course-to-compass-hea/96346#96346

@sophit since it takes the direction from which the wind is coming, there is an "implied" 180° rotation which turns it into a subtraction, which is what I need. Ground velocity = air velocity + wind velocity, so air velocity= ground velocity - wind velocity.

@fab that does the opposite of what I need. I'm trying to find the airspeed of a model plane given the speed and direction of the plane from GPS and the wind speed and direction from an external source.

Yes you would have to invert that equation.

Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [meta], or in [chat]. Comments continuing discussion may be removed.

@DeltaLima (just commenting to ping you) I posted a question on Meta: https://aviation.meta.stackexchange.com/questions/4416/is-there-an-app-for-at-least-chat Because I'm on my phone, I can't use chat.

@kj7rrv Whilst the UI is not very nicely adapted for mobile use, you can use chat on a phone.

@DeltaLima not on my phone; the text box disappears as soon as I tap it.

If you apply the law of cosines to the triangle formed by the vectors $\vec{A}$ (Airspeed), $\vec{W}$ (Wind) and $\vec{G_S}$ (Ground speed), where $\vec{G_S}=\vec{A}+\vec{W}$, you get:
$$|\vec{A}|=\sqrt{|\vec{G_S}|^2+|\vec{W}|^2-2|\vec{G_S}|\cdot|\vec{W}|\cdot\mathrm{cos}(\theta)}\tag{1}$$
where $\theta=\theta_{G_S}-\theta_W$.
As you yourself pointed out in the comments below, wind directions are usually reported as the direction winds originate from, not where the blow to. With that convention, the angle $\theta_W$ should be expressed as $\theta_W=\theta_W'+180$. It's easy to show that the additional $180$ can be brought out the cosine as a "-" minus sign. The final equation is therefore:
$$|\vec{A}|=\sqrt{|\vec{G_S}|^2+|\vec{W}|^2+2|\vec{G_S}|\cdot|\vec{W}|\cdot\mathrm{cos}(\theta)}\tag{2}$$
That's equivalent to yours.

My code takes the direction from which the wind is coming, so it's already negated by that. Since velocity is a vector, IIUC either speed or direction needs to be negated, and taking direction in the opposite direction suffices. Is that correct? In testing, my code seems to work.

"your equation has a sign that's incorrect: it should be −VWind⋅cos(θ)
, not +VWind⋅cos(θ)": I've already discussed this issue with @kj7rrv, but apparently kj7rrv prefers to have +Vwind.

@sophit it's not that I prefer to have +Vwind; it's that I prefer to take the wind direction in the way it's normally reported, which requires using +Vwind.

Ah yes, if you want to use that convention (and yes you're right, you should), then $\theta$ must be computed differently. But eventually the sign will come out of the cosine and you'll get the equation you have. I'll edit my answer to reflect this.


text:
What are some categories of accidents other than takeoff and landing?
From the FAA Remote Pilot study guide, but apparently referring to manned aircraft accidents: "...the vast majority of these accidents occur during landing (24.1 percent) and takeoff (23.4 percent)."
24.1 + 23.4 = 47.5, a minority, but this could still be a "vast majority" if there are a lot of small categories making up the majority of accidents. Assuming this sentence is not wrong, there must be at least three other categories, but for it to really be a "vast majority" more than three would be needed.
The only other category I can think of is cruise flight, but I'm not sure that would be considered a category rather than broken into more specific phases. What are some categories of accidents other than takeoff and landing?

Maybe those other categories are climb, approach, cruise and manoeuvring - but I'm not sure. Please share that FAA reference :-)

@AdityaSharma https://www.faa.gov/regulations_policies/handbooks_manuals/aviation/media/remote_pilot_study_guide.pdf it's on page number 50 (59 in the PDF).

I would suggest you edit your title to read " What are some categories of accidents other than takeoff, landing, and cruising?", which is what you asked at the end.  (Or delete "and cruising" from the final sentence.)  As it stands now things don't match up. 2)  "Controlled flight into terrain" is an obvious choice that would rarely fit with "take-off" or landing but certainly might fit with "cruising".  CFIT accidents are rather common.  3) Other possibilities are engine failure, or loss of control in cloud.

But your question is fundamentally about phases of flight, not causes of accidents.
Ok, I get that.  But I would submit that if someone states "the vast majority of accidents happen during X  + Y phases", then that means a vast majority of the total accidents. It doesn't mean that the sum of accident in X+Y phases vastly outnumbers the number of accidents in any one other phase.

So, it doesn't really matter into how many pieces you slice up the remaining phases of flights  experiencing accidents.  Question could perhaps be edited--

@quietflyer that's what I always thought "vast majority" meant too, but I asked on a non-aviation site just about the logic of this, and someone else said that "vast majority" means that all other categories are individually insignificant, even if collectively they are a majority.

It's not completely cut-and-dried-- thanks for considering my thoughts.

The official "Phase of Flight" list that FAA accident reports use can be found here in the aidcodes.doc file and includes the following:
TABLE ID : PHS      DESCRIPTION : PHASE OF FLIGHT

CODE                LONG DESCRIPTION                            SHORT DESCRIPTION
__________      ______________________________              ____________________

AA              STARTING ENGINES                                STATIC-STARTUP
AB              IDLING ENGINES                                  STATIC-IDLE ENG
AC              ENGINE RUN-UP                                   STATIC-RUN UP
AD              IDLING ROTORS                                   STATIC-IDLE ROTOR
BA              POWER ON DESCENT (ROTORCRAFT)           APPR-ROTOR RWR DSC
BB              AUTOROT DESCENT (ROTORCRAFT)            APPR-AUTOROT DES
BC              IN TRAFFIC PATTERN-CIRCLING                 APPR-VFR PTRN CIRC
BD              FINAL APPROACH                                  APPR-FINAL
BE              INITIAL APPROACH-IFR                        APPR-INITIAL
BF              FINAL APPROACH-IFR                              APPR-FINAL
BG              GO ROUND (ABORTED VFR)                      APPR-GO ARND VFR
BH              MISSED APPROACH IFR                         APPR-MISSED IFR
BI              FORCED LANDING                                  APPR-FORCE LAND
CX              CLIMB TO CRUISE                                 CLIMB-TO CRUISE
DA              NORMAL CRUISE                                   CRUISE-NORMAL
DB              HOLDING IFR                                     CRUISE-HOLD IFR
DC              ACROBATICS-BUZZING                          CRUISE-ACRO/BUZZ
DD              FCD/PREC LDG FROM CRUISE                    CRUISE-FCD/PREC LD
DE              LOW LEVEL OPERATIONS                        CRUISE-LO LVL OF
DF              UNAUTH LOW LEVEL OPNS                       UNAUTH LOW LEVEL
DG              PRACTICE TRAINING MANUEVERS                 PRACTICE MANUEVER
DH              FORMATION FLYING                                FORMATION
EX              DESCENT                                         DESCENT
FA              OPS ON GRND TO TKOF/RTRCRFT                 GRND ROTOR/GR-TO
FB              OPS ON GRND FROM LDG/RTRCRFT                GRND FRM LAND
FC              AERIAL TAXI TO TKOF/RTRCRFT                 GRND ROTOR/AIR-TO
FD              AERIAL TAXI FROM LDG/RTRCRFT                GRND ROTOR/AIR-LND
FE              GROUND TAXI, OTHER AIRPLANE                 GRND GRD TAXI
FF              AERIAL TAXI, OTHER HELICOPTER               GRND AER TAXI
FG              OTHER GROUND OPS                                GRND OTHER
GX              HOVERING                                        HOVERING
HA              LEVEL OFF TOUCHDOWN                         LAND-LVL OFF TOUCH
HB              ROLL-OUT (FIXED WING)                       LAND-ROLL FW
HC              ROLL-OUT (ROTORCRAFT)                       LAND-ROLL HELIO
HD              PWR ON VERTICAL LDNG/RTRCRFT            LAND-PWR ON VERT
HE              PWR OFF VERT LAND/AUTOROTATE            LAND-PWR OFF VERT
HF              PRW ON RUN LANDING (ROTORCRAFT          LAND-PWR ON RUN
HG              PWR OFF RUN LAND (AUTOROTATION          LAND-PWR OFF RUN
HH              TOUCH AND GO LANDING                        TOUCH AND GO
HK              SLOPE LANDING                                   SLOPE
HM              SETTLE WITH POWER (ROTORCRAFT)          SETTLEWPWR
IA              TAKEOFF GROUND ROLL                         TKOF-GRND ROLL
IB              TO INIT CLIMB(1ST PWR REDUCT)               TKOF-INIT CLIMB
IC              TO VERTICAL (HELICOPTER ONLY)               TKOF-VERTICAL
ID              TO RUNNING(HELICOPTER/VTOL AC)          TKOF-RUNNING
IE              TO ABORTED (FIXED WING)                     TKOF-ABORTED
IF              TKOF-ABORT-COPTER VERTICAL                  TKOF-ABORT-VERT
IG              TO ABORTED (COPTER/RUN TAKOFF)          TKOF-ABORT RUNNING
IH              FORCED/PRECAUTIONARY LANDING            TKOF-FORCE LAND
JX              UNKNOWN                                         UNKNOWN
KA              SIMULATED FORCED LANDING/TKOF           SFL-TKOF RUN
KB              SFL TAKEOFF CLIMB                               SFL-TKOF CLIMB
KC              SFL CRUISE                                      SFL-CRUISE
KD              SFL FINAL APPROACH                              SFL-FNL APP
KE              SFL TRAFFIC PATTERN                             SFL-TFC PTRN
KF               SFL GO AROUND                                  SFL-GO-ARND
KG              SFL AUTOROTATION                                SFL-AUTO ROT
PJ              PARACHUTE JUMPING                           PARA JUMP
PL              PINNACLE LANDING                                PINNACLE
SR              SCUD RUNNING                                    SCUD
99999           PHASE OF FLIGHT INVALID                     ERROR MESSAGE

Reports like this from a third party source roll the data up into some condensed categories but the above list is the official list used by the NTSB for Aviation reports.
Note: you can pull the raw NTSB accident data in spreadsheet form from the link above (its year by year but you can still get it) and roll up your own stats if you are indeed curious.